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Theorem recexprlemex 7457
Description: 𝐵 is the reciprocal of 𝐴. Lemma for recexpr 7458. (Contributed by Jim Kingdon, 27-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
Assertion
Ref Expression
recexprlemex (𝐴P → (𝐴 ·P 𝐵) = 1P)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Proof of Theorem recexprlemex
StepHypRef Expression
1 recexpr.1 . . . 4 𝐵 = ⟨{𝑥 ∣ ∃𝑦(𝑥 <Q 𝑦 ∧ (*Q𝑦) ∈ (2nd𝐴))}, {𝑥 ∣ ∃𝑦(𝑦 <Q 𝑥 ∧ (*Q𝑦) ∈ (1st𝐴))}⟩
21recexprlemss1l 7455 . . 3 (𝐴P → (1st ‘(𝐴 ·P 𝐵)) ⊆ (1st ‘1P))
31recexprlem1ssl 7453 . . 3 (𝐴P → (1st ‘1P) ⊆ (1st ‘(𝐴 ·P 𝐵)))
42, 3eqssd 3114 . 2 (𝐴P → (1st ‘(𝐴 ·P 𝐵)) = (1st ‘1P))
51recexprlemss1u 7456 . . 3 (𝐴P → (2nd ‘(𝐴 ·P 𝐵)) ⊆ (2nd ‘1P))
61recexprlem1ssu 7454 . . 3 (𝐴P → (2nd ‘1P) ⊆ (2nd ‘(𝐴 ·P 𝐵)))
75, 6eqssd 3114 . 2 (𝐴P → (2nd ‘(𝐴 ·P 𝐵)) = (2nd ‘1P))
81recexprlempr 7452 . . . 4 (𝐴P𝐵P)
9 mulclpr 7392 . . . 4 ((𝐴P𝐵P) → (𝐴 ·P 𝐵) ∈ P)
108, 9mpdan 417 . . 3 (𝐴P → (𝐴 ·P 𝐵) ∈ P)
11 1pr 7374 . . 3 1PP
12 preqlu 7292 . . 3 (((𝐴 ·P 𝐵) ∈ P ∧ 1PP) → ((𝐴 ·P 𝐵) = 1P ↔ ((1st ‘(𝐴 ·P 𝐵)) = (1st ‘1P) ∧ (2nd ‘(𝐴 ·P 𝐵)) = (2nd ‘1P))))
1310, 11, 12sylancl 409 . 2 (𝐴P → ((𝐴 ·P 𝐵) = 1P ↔ ((1st ‘(𝐴 ·P 𝐵)) = (1st ‘1P) ∧ (2nd ‘(𝐴 ·P 𝐵)) = (2nd ‘1P))))
144, 7, 13mpbir2and 928 1 (𝐴P → (𝐴 ·P 𝐵) = 1P)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104   = wceq 1331  wex 1468  wcel 1480  {cab 2125  cop 3530   class class class wbr 3929  cfv 5123  (class class class)co 5774  1st c1st 6036  2nd c2nd 6037  *Qcrq 7104   <Q cltq 7105  Pcnp 7111  1Pc1p 7112   ·P cmp 7114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-eprel 4211  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-1o 6313  df-2o 6314  df-oadd 6317  df-omul 6318  df-er 6429  df-ec 6431  df-qs 6435  df-ni 7124  df-pli 7125  df-mi 7126  df-lti 7127  df-plpq 7164  df-mpq 7165  df-enq 7167  df-nqqs 7168  df-plqqs 7169  df-mqqs 7170  df-1nqqs 7171  df-rq 7172  df-ltnqqs 7173  df-enq0 7244  df-nq0 7245  df-0nq0 7246  df-plq0 7247  df-mq0 7248  df-inp 7286  df-i1p 7287  df-imp 7289
This theorem is referenced by:  recexpr  7458
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